| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Optimization with constraint |
| Difficulty | Standard +0.3 This is a standard optimization problem requiring expressing surface area in terms of one variable using a volume constraint, then finding and classifying a stationary point using basic differentiation. The algebra is straightforward and the method is routine for A-level, making it slightly easier than average. |
| Spec | 1.02z Models in context: use functions in modelling1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(3x^2y = 288\) \(y\) is the height | B1 | co |
| \(A = 2(3x^2 + xy + 3xy)\) | M1 | Considers at least 5 faces (\(y \neq x\)) |
| Sub for \(y \rightarrow A = 6x^2 + \frac{768}{x}\) | A1 | co answer given |
| [3] | ||
| (ii) \(\frac{dA}{dv} = 12x - \frac{768}{x^2}\) | B1 | co |
| \(= 0\) when \(x = 4 \rightarrow A = 288\). Allow \((4, 288)\) | M1 A1 | Sets differential to 0 + solution. co |
| \(\frac{d^2A}{dx^2} = 12 + \frac{1536}{x^3}\) | M1 | Any valid method |
| \((= 36) > 0\) Minimum | A1 | co www dep on correct f" and \(x = 4\) |
| [5] |
(i) $3x^2y = 288$ $y$ is the height | B1 | co |
$A = 2(3x^2 + xy + 3xy)$ | M1 | Considers at least 5 faces ($y \neq x$) |
Sub for $y \rightarrow A = 6x^2 + \frac{768}{x}$ | A1 | co answer given |
| | [3] |
(ii) $\frac{dA}{dv} = 12x - \frac{768}{x^2}$ | B1 | co |
$= 0$ when $x = 4 \rightarrow A = 288$. Allow $(4, 288)$ | M1 A1 | Sets differential to 0 + solution. co |
$\frac{d^2A}{dx^2} = 12 + \frac{1536}{x^3}$ | M1 | Any valid method |
$(= 36) > 0$ Minimum | A1 | co www dep on correct f" and $x = 4$ |
| | [5] |9 The base of a cuboid has sides of length $x \mathrm {~cm}$ and $3 x \mathrm {~cm}$. The volume of the cuboid is $288 \mathrm {~cm} ^ { 3 }$.\\
(i) Show that the total surface area of the cuboid, $A \mathrm {~cm} ^ { 2 }$, is given by
$$A = 6 x ^ { 2 } + \frac { 768 } { x }$$
(ii) Given that $x$ can vary, find the stationary value of $A$ and determine its nature.
\hfill \mbox{\textit{CAIE P1 2014 Q9 [8]}}